Linear Optimization

Linear optimization, also known as linear programming (LP), is a mathematical method used to find the best possible outcome (like maximum profit or minimum cost) in a model where the objective and the requirements (constraints) are represented by linear relationships.[1] 🎯

Think of it as a way to make the smartest decisions when you have limited resources and a clear goal.

Here’s a breakdown of its key components:

• Objective Function: This is a linear mathematical equation that defines what you want to achieve.[2] It’s the quantity you aim to maximize (e.g., profit, output) or minimize (e.g., cost, waste).[3] For example, if a company makes two products, A and B, with profits of $10and$12 respectively, the objective function to maximize profit (P) would be: $P=10A+12B$.
• Decision Variables: These are the unknown quantities that you need to determine to reach your objective.[4] In the example above, the number of units of product A and product B to produce are the decision variables.[5]
• Constraints: These are linear inequalities or equalities that represent the limitations or restrictions on the decision variables.[6] These could be resource availability (like raw materials, labor hours, or machine capacity), production limits, or demand restrictions.[7] For instance, if producing product A requires 2 hours of labor and product B requires 3 hours, and there are only 100 labor hours available, a constraint would be: $2A+3B\le 100$.
• Non-negativity: In most real-world scenarios, decision variables cannot be negative (e.g., you can’t produce a negative number of products).[8] So, decision variables are usually constrained to be greater than or equal to zero.[9] In this case: $A\ge 0$, $B\ge 0$.

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Key Characteristics

• Linearity: The relationships between the decision variables in both the objective function and the constraints must be linear. This means no exponents, square roots, or multiplying variables together.
• Finiteness: There must be a finite number of decision variables and constraints.[10]
• Certainty: The parameters in the objective function and constraints (like profit per unit or resource availability) are assumed to be known and constant.
• Divisibility: Decision variables can often take on fractional values, though a related field called integer programming deals with variables that must be whole numbers.[11]

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How is it Solved?

Several methods are used to solve linear optimization problems:

• Graphical Method: This visual approach can be used for problems with only two decision variables.12 Constraints are plotted on a graph to identify a “feasible region” (the area where all constraints are satisfied).[13] The optimal solution is typically found at one of the corner points of this feasible region.[14]
• Simplex Method: This is a more general and widely used algebraic algorithm that iteratively moves from one corner point of the feasible region to an adjacent one, improving the objective function value at each step until the optimal solution is found.[15]
• Interior-Point Methods: These are another class of algorithms that can be very efficient for large-scale linear programming problems.[16] They approach the optimal solution from the interior of the feasible region.[17]

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Why is it Important? 💡

Linear optimization is a powerful tool used across many fields due to its ability to handle complex problems and provide optimal solutions.[18] Common applications include:

• Business and Finance: Optimizing investment portfolios, production planning, resource allocation, and supply chain management.[19]
• Manufacturing: Minimizing production costs, maximizing output, and scheduling.[20]
• Logistics and Transportation: Finding the most efficient routes for delivery, scheduling shipments, and managing inventory.[21]
• Agriculture: Determining the best crop mix to plant given land, water, and labor constraints.[22]
• Telecommunications: Optimizing network flows.[23]
• Energy Sector: Managing power generation and distribution.[24]

In essence, linear optimization provides a structured and mathematical way to make the best possible decisions in situations with multiple choices and limitations.[25]

Example

Simple Profit Maximization Example

Gist: https://gist.github.com/prasanth-ntu/eb5689334b9aa72410e4714f3d171761

Maximizing Profit DP Example

Google Colab: https://colab.research.google.com/drive/1A6TqIPzS_2zohu0dQojWi2sybg39MaMY?usp=sharing